New representation of n-mode squeezed state gained via n-partite entangled state

New representation of n-mode squeezed state gained via n-partite entangled state

Optics Communications 254 (2005) 256–261 www.elsevier.com/locate/optcom New representation of n-mode squeezed state gained via n-partite entangled st...

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Optics Communications 254 (2005) 256–261 www.elsevier.com/locate/optcom

New representation of n-mode squeezed state gained via n-partite entangled state q Nian-Quan Jiang Department of Physics, Wenzhou Normal College, Wenzhou 325027, China Received 30 January 2005; received in revised form 27 March 2005; accepted 25 May 2005

Abstract By virtue of the n-partite entangled state, we extend the way of Agarwal–SimonÕs presenting single-mode squeezed state to n-mode case and find a new representation of the n-mode squeezed state. This n-mode squeezed state is also an entangled state and can be a superposition of n-mode coherent states.  2005 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.67.a; 03.65.Bz Keywords: n-Mode squeezed state; Entangled state; New representation of squeezing

Squeezed states have been important topic since 1970s due to their wide applications in optical communication and precise measurement in quantum optics [1]. They exhibit interesting nonclassical behavior. Many attempts have been made to find new squeezed states and new form of squeezing operators so that new experimental implementation could be proposed [2,3]. For the single-mode squeezed state wave func1=4 tion Wðx; rx Þ ¼ ½1=ðpr2x Þ  expðx2 =2r2x Þ (its wave packetÕs width is r2x =2) Agarwal and Simon found a new representation [4]   1 2 2 ðs jWix  s1=2 exp   1ÞX ð1Þ s2x ¼ 1=r2x ; x 1 j0i 2 x 1=4

or for Wðp; rp Þ ¼ ½1=ðpr2p Þ  expðp2 =2r2p Þ,   1 jWip  sp1=2 exp  ðs2p  1ÞP 21 j0i s2p ¼ 1=r2p ; 2 q

Work supported by the National Natural Science Foundation of China under Grant 10447128. E-mail address: [email protected]

0030-4018/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.05.046

ð2Þ

N.-Q. Jiang / Optics Communications 254 (2005) 256–261

where X1 and P1 are the coordinates and momentum operators, respectively. Let f ¼  12 ðs2p  1Þ we have the single-mode squeezing operator, S 1  ð1  2f Þ

1=2 fP 2 1

e

1=2 fP 2 1

e

sp ¼ ek , ð3Þ

.

Using S1, Eq. (2) can be expressed as jWip  ð1  2f Þ

257

j0i ¼ sech1=2 k exp



 1 y2 a1 tanh k j0i; 2

ð4Þ

which is just the traditional standard form of the single-mode squeezed vacuum state. Moreover, Agarwal– SimonÕs representation shows that the single-mode squeezed state can be a superposition of coherent states on a line in phase space, i.e., !1=2 Z ! 1 sp x2 jWip ¼ dx exp  2 ð5Þ jxicoh ; pðs2p  1Þ sp  1 1 pffiffiffi where jxicoh ¼ expði 2P 1 xÞj0i is coherent state [5,6]. Enlightened by Agarwal–SimonÕs idea and employing the two-mode entangled state, in [7] we have derived a new form of the two-mode squeezing operator S 2  ð1  2f Þ

1=2 f ½ðX 1 X 2 Þ2 þðP 1 þP 2 Þ2 =2

;

ð6Þ

    1 f  y : exp a1  a2 a1  ay2 . 1f 1f

ð7Þ

e

whose normally product form is S 2 ¼ ð1  2f Þ

1=2

Operating S2 on the vacuum state |00æ will yield the two-mode squeezed state S 2 j00i ¼ sechk exp½ay1 ay2 tanh kj00i;

ð8Þ

which is just the traditional standard form of the two mode squeezed vacuum state. An interesting problem thus naturally arises: Can Agarwal–SimonÕs idea be extended to n-mode case? We will deal with this problem in the article. Firstly, we need introduce a n-mode entangled state with continuous variables. To derive compact result, we choose the Jacobian coordinates used in traditional many-body theory [8] as observable variables (One of the advantages of using Jacobian coordinates is that the motion of center-of-mass can be separated off from the relativeP motion of many particles). Noting that for n-partite system Pnthe Jacobian j1 1 coordinates operators X j  j1 X ðj ¼ 2; . . . ; nÞ and the total momentum operator k k¼1 i¼1 P i are permutable with each other, we can introduce the common eigenstate of them [9] " pffiffiffi n n n X pffiffiffi X 1 1 1 2 2ip X 2 ðj  1Þvj  p þ ayj þ 2 vj ayj jpv2 v3 . . . vn i ¼ pffiffiffi n=4 exp  2j 2n np n j¼2 j¼1 j¼1 ! #  X j n n n X X pffiffiffi X vj 2 1 1   2 ayk þ ayj ayk þ ay2 j0    0i; j n n 2 j k¼1 j¼1 jk;j;k¼1 j¼1 where v1 = 0 which obeys the eigenvector equations ! n X P i jpv2 v3 . . . vn i ¼ pjpv2 v3 . . . vn i i¼1

and

ð9Þ

ð10Þ

258

N.-Q. Jiang / Optics Communications 254 (2005) 256–261

!

Xj 

j1 1 X X k jpv2 v3    vn i ¼ vj jpv2 v3 . . . vn i j  1 k¼1

j ¼ 2; . . . ; n.

The Schmidt decomposition of |pv2v3. . .vnæ in n-mode coordinates basis is ! n X 1 vk jpv2 v3 . . . vn i ¼ pffiffiffiffiffiffi exp ip ; k 2p k¼2 Z 1 n1 X 1 v i expðixpÞ; dxjxi1  jx þ v2 i2      jx þ vn þ j j n 1 j¼2

ð11Þ

ð12Þ

where |xæi (i = 1,2,. . .,n) is the coordinates eigenstate of Xi, Xi|xæi = x|xæi 1=4

jxii ¼ p



 1 2 pffiffiffi y 1 y2 exp  x þ 2xai  ai j0ii . 2 2

ð13Þ

According to the standard theory of Schmidt decomposition of an entangled state [10], we know that |pv2v3. . .vnæ is an entangled state. Using the technique of integral within an ordered product (IWOP) of operators [11] we can directly prove |pv2. . .vnæ span a complete set Z

1

dp dv2 . . . dvn jp; v2 ; . . . ; vn ihp; v2 ; . . . ; vn j ¼ 1.

ð14Þ

1

Specially, when n = 3, Eq. (9) reduces to " pffiffiffi pffiffiffi  pffiffiffi    3 1 1 2 1 2 1 2 2 X 2 3 2 3 y y v2 þ v3 a1 þ v2  v3 ay2 ip jpv2 v3 i ¼ pffiffiffi exp  v2  v3  p þ aj  4 3 6 3 3 2 3 2 3p3=4 j¼1 # pffiffiffi 3 3 2 2 2 X 1X v3 ay3 þ þ ayj ayk  ay2 j0    0i; ð15Þ 3 3 jk;j;k¼1 6 j¼1 j which obeys ðP 1 þ P 2 þ P 3 Þjpv2 v3 i ¼ pjpv2 v3 i; ðX 2  X 1 Þjpv2 v3 i ¼ v2 jpv2 v3 i

ð16Þ ð17Þ

and 

 1 X 3  ðX 1 þ X 2 Þ jpv2 v3 i ¼ v3 jpv2 v3 i; 2

If we let v2 ! y 2 ; v3 ! 12 y 2  y 3 ; then |pv2v3æ becomes pffiffiffi 3  2  i 2p X 1 p 1 2 2 exp   y 2 þ y 3  y 2 y 3 þ ay jpy 2 y 3 i ¼ pffiffiffi 3 i¼1 i 6 3 3p3=4 # pffiffiffi pffiffiffi 3 3   2X 1X 2y 2  y 2y 3  y y y y y y y y2 a1  2a2 þ a3 þ a1 þ a2  2a3 þ þ aa  a j000i. 3 i
ð18Þ

ð19Þ

N.-Q. Jiang / Optics Communications 254 (2005) 256–261

Corresponding to (3) and (6), we introduce n-mode squeezing operator 8 2 !2 !2 39 j1 n n < X = X X j  1 1 1 n=4 Xj  S n  ð1  2f Þ exp f 4 Xk þ Pi 5 . : j¼2 j ; j  1 k¼1 n i¼1

259

ð20Þ

Using (10), (11) andP(14) as well as the normally ordered form of the vacuum projector j0    0ih0    0j ¼: exp½ ni¼1 ayi ai :, we can derive the normal product form of Sn, " !# Z 1 n X j1 2 1 2 n=4 vj þ p S n ¼ ð1  2f Þ dp dv2 . . . dvn  exp f jp; v2 ; . . . ; vn ihp; v2 ; . . . ; vn j j n 1 j¼2   n=4  ð1  2f Þ f n K ¼ : exp þ K þ 2K  ; ð21Þ þ  0 n=2 ð1  f Þ 2 ð1  f Þ where Kþ 

n n 1X 2 2 X ð1  Þay2  ay ay ; 2 i¼1 n i n i
K 

K yþ ; 2K 0



n X i¼1

ayi ai

ð22Þ

n þ ; 2

compose a closed SU(1,1) Lie algebra, ½K 0 ; K þ  ¼ K þ ; ½K 0 ; K   ¼ K  ; ½K  ; K þ  ¼ 2K 0 . Operating (21) on |0  0æ we derive   n=4

ð1  2f Þ f K þ j0    0i ¼ sechn=2 k exp½K þ tanh k 0    0i; S n j0    0i ¼ exp n=2 1f ð1  f Þ s2 1

ð23Þ

ð24Þ

2s

p where f ¼ 12 ð1  s2p Þ , sp = ek, tanh k ¼ sp2 þ1 ; sechk ¼ s2 þ1 , which is the standard form of the n-mode squeezed p p vacuum state. Calculating the quantum fluctuation of the two photon-field quadratures

3 3 1 X 1 X P i ; X qua ¼ pffiffiffi X i; P qua ¼ pffiffiffi n i¼1 n i¼1

ð25Þ

in the state Sn|0  0æ, we obtain  2  2 1 1 h 4X qua i ¼ s2p ; h 4P qua i ¼ s2 ; 4 4 p which show the standard squeezing. Thus, 8 2 !2 !2 39 j1 n n < X = X X j  1 1 1 n=4 Xj  Xk þ P i 5 j0    0i; ð1  2f Þ exp f 4 : j¼2 j ; j  1 k¼1 n i¼1

ð26Þ

ð27Þ

is the new representation of n-mode squeezed vacuum state in the sense of Agarwal and Simon. From (27) we see that for a n-partite system once the Jacobian coordinates and total momentum are chosen as observable variables, n-mode squeezing operator is just the form similar to (3) which was introduced by Agarwal and Simon, so the Jacobian coordinates exhibit some symmetry of n-partite system. Especially, when n = 2 (20) reduce to (6); when n = 3 (20) becomes,

260

N.-Q. Jiang / Optics Communications 254 (2005) 256–261

8 2 !2 39  2 3 < 1 = X 2 1 1 3=4 2 X 3  ðX 1 þ X 2 Þ þ S 3  ð1  2f Þ exp f 4 ðX 2  X 1 Þ þ Pi 5 . : 2 ; 3 2 3 i¼1

ð28Þ

Moreover, the form (27) provides us a new view for the construction of the n-mode squeezed states, due to

"

# j1 i1 1 X 1 X X k; X i  X k ¼ 0; Xj  j  1 k¼1 i  1 k¼1 " # j1 n X 1 X X k; P i ¼ 0; i; j ¼ 2; 3;    ; n; Xj  j  1 k¼1 i¼1

ð29Þ

we have S n j0    0i ¼

nð1  2f Þn=4 n=2

ð4f pÞ "

Z

Z

1

1

du 1

dv2 . . . dvn 1 n X

!#

 1 X n n1

n1 P



vn Xk 1 j 2 k¼1 nu2 þ vj euP ev2 ðX 2 X 1 Þ    e j0    0i; ð30Þ 4f j1 j¼2 Pn1 1 where euP ev2 ðX 2 X 1 Þ    evn ðX n n1 k¼1 X k Þ j0    0i is a n-mode coherent state. Eq. (30) indicates that the n-mode squeezed state can be a superposition of n-mode coherent states only along a line in the complex plane of each mode. In summary, we have extended the new representation of squeezed state in the sense of Agarwal and Simon to the n-mode case, in our discussion the n-partite entangled state |pv2. . .vnæ plays an essential role. The new representation implies that the n-mode squeezed state can also be a superposition of coherent states. Because the n-mode squeezed vacuum state Sn|0  0æ is also entangled state, they may have potential applications in quantum information.

 exp 

Acknowledgement Thank referees for their constructive comments.

References [1] See, e.g., G.M. DÕAriano, M.G. Rassetti, J. Katriel, A.I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light, Plenum, New York, 1989, p. 301; V. Buzˇek, J. Mod. Opt. 37 (1990) 303; R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709. [2] See, e.g., L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge, 1995. [3] V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1. [4] G.S. Agarwal, R. Simon, Opt. Commun. 92 (1992) 105. [5] R.J. Glauber, Phys. Rev. 130 (1963) 2529; R.J. Glauber, Phys. Rev. 131 (1963) 2766. [6] J.R. Klauder, B.S. Skargerstam, Coherent States, World Scientific, Singapore, 1985. [7] H.Y. Fan, N.Q. Jiang, H.L. Lu, Opt. Commun. 234 (2004) 277. [8] D.I. Blokhintsev, Quantum Mechanics, Reidel, Dordrecht, 1964. [9] N.-Q. Jiang, Phys. Lett. A 339 (2005) 255. [10] J. Preskill, Physics Lectures, California Institute of Technology, Pasadena, 1998, p. 229.

N.-Q. Jiang / Optics Communications 254 (2005) 256–261 [11] H.Y. Fan, H.R. Zaidi, J.R. Klauder, Phys. Rev. D 35 (1987) 1831; H.Y. Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147. [12] H.Y. Fan, N.Q. Jiang, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) 283.

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